The Czechs were up 2-0 after only six minutes, leaving Greece too much to do to get back in the match.
"It was a game that started really badly for us," coach Fernando Santos, whose team suffered only their second defeat in 23 matches since the 2010 World Cup, told reporters.
"We knew they would pressure us and we failed to respond. Unfortunately the first 10 minutes really cost us. We did everything we could to score the equaliser but we couldn't."
A makeshift central defence with Kyriakos Papadopoulos and Kostas Katsouranis had no time to adjust as the Czechs grabbed a third-minute lead with Petr Jiracek before Vaclav Pilar added another three minutes later.
"This is a line-up that delivered a very good second half against the Polish team [in a 1-1 draw]. That is why I chose this line-up," said Santos.
The Greeks cut the deficit early in the second half through substitute Fanis Gekas but failed to find another goal despite having the upper hand for the remainder of their game.
"In the last 15 minutes we used our hearts more than our minds and this clearly didn't pay off," said Santos.
They face Russia, who play Poland later on Tuesday, in their last group match.
"We paid dearly for our very bad start," said midfielder Giorgos Samaras. "We must now go into that last game and grab any chances we have left in the tournament. We wanted the win today. We still have faith in our team and ourselves."
Scorer Fanis Gekas said the group was still open.
"This third game is a final for us," he said. "The group is still open and everyone can advance so we won't give up. We will keep at it."
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